Absolute Value Solver and Calculator

This absolute value solver solves equations such as |2x − 3| = 7 by splitting them into two cases, checks each answer, and tells you when there are no solutions. It also finds the absolute value of a single number, such as |−7.5| = 7.5.

Result

Enter values and press Calculate to see the result.

Formula

∣u∣=c  (c>0)  ⟺  u=c or u=−c\lvert u \rvert = c \;(c > 0) \iff u = c \text{ or } u = -c

Split the equation into two cases. If c = 0 there is one case; if c < 0 there is no solution.

∣x∣={xx≥0−xx<0\lvert x \rvert = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}

The absolute value is the distance from 0, so it is never negative.

How to use this calculator

  1. To solve an equation, choose “Solve |ax + b| = c” and enter a, b and c — for |2x − 3| = 7 that is 2, −3 and 7.
  2. To find the absolute value of a number, choose the second option and enter the number: drop its minus sign, if it has one.
  3. Press Calculate to see both cases, the solutions and the check.

Worked examples

Solve |2x − 3| = 7

Case 1: 2x − 3 = 7, so x = 5. Case 2: 2x − 3 = −7, so x = −2.

Check: |2·5 − 3| = 7 and |2·(−2) − 3| = |−7| = 7.

One solution: |x + 4| = 0

Only 0 has absolute value 0, so x + 4 = 0 and x = −4.

No solution: |3x − 1| = −2

An absolute value is never negative, so it can’t equal −2: no solution.

The absolute value of −7.5

−7.5 is 7.5 units from 0, so |−7.5| = 7.5.

Common mistakes

  • Solving only the positive case: |u| = 7 also gives u = −7.
  • Solving |u| = −2 anyway — an absolute value can’t be negative.
  • Distributing the bars: |a − b| is not |a| − |b|.
  • Forgetting to check answers, which matters when x appears outside the bars too.

Frequently asked questions

How do you find the absolute value of a number?

Take its distance from 0: remove the minus sign if there is one. |−12| = 12 and |5| = 5.

How do you solve an absolute value equation?

Get the absolute value on its own, then split into two equations: the inside equals c, or the inside equals −c. Solve both and check.

When does an absolute value equation have no solution?

When the absolute value is set equal to a negative number, such as |3x − 1| = −2.

What about absolute value inequalities?

Use the inequality calculator: it solves |2x − 1| ≥ 5 and similar, and draws the solution on a number line.